Grissim, my main suggestion is to cut way back or eliminate the handstand training for a while. Next, when you are doing handstands, you very much need to "squeeze" the heck out of the ground with the tips of all of your fingers while trying to keep the base of your index finger in contact with the ground. Loose fingers and wrists are a quick way to injury in handstands.

Take care of wrist/hand problems early, or end up with potentially chronic issues that affect all of your life activities later (since we humans use our hands for just a *few* things).

Also note that carpal tunnel syndrome-ish symptoms can also originate from the 1st rib / clavicular area or directly from the cervical spine. If you have had any tweaks to either of those areas recently, you might want to find a professional to sort that out.

Dr. G ain't kidding, I know this first hand... (oops no pun intended)
In addition to what everyone said, shoulder/scapular mobility is super important if you plan to get on your hands again. I'd definitely invest time in increasing your ROM there. Unfortunately it's not as simple as doing lots of shoulder stretches. An adult will need to something that gives you better movement awareness of the area. Or at least that's been my experience. Feldenkrais is what worked for me but there are a lot of methods out there that are good.